Guzzler, think about it. How could we possibly be excreting fat in our breath? Gaseous fat?? Globules of fat? How would fat get out of the bloodstream and into the lungs???
When fat is used by the body, it creates ketones, and these in turn become acetyl molecules, and it is these that get into our breath, giving it the characteristic pear drop aroma of acetone. (keto beath)
In addition, some of the excess fat gets converted into bile, and gets exctreted in the poop via the bile duct. This fat passing into the large intestine helps improve motility and helps overcome constipation, so I am led to believe. So yes, excretion is an apt word to use,
However it is the burning of the fat stores by which process the complex fat molecules get broken down into the acetyl CoA components that our body's furnace uses as fuel. CoA is generated by both carbs via glucogen formation, or by converting the lipid fats in the abdomen into CoA. Same fuel in the end, but our bodies only convert fat to CoA while the glucose levels are low since this is the backup process, It takes more energy to convert fat to fuel, and our bodies only do this as a last resort.
It needs certain enzymes to make the process work properly, and these are created during the fat adaption process. Once the enzymes have been created by our bodies, then switching between glucose and fat as a fuel source becomes automatic, and depends on bgl levels and adrenalin and insulin levels that combine to control the conversion, It is not a simple process, but is one which our bodies are designed to do naturally.
There is another set of enzymes that control the conversion of fat into useful stuff while being digested. These enzymes work to limit how much fat is converted, and so not all fat is passed into the cholesterol and lipid pathway. It just gets excreted like other waste products. So our body naturally limits how much of the fat is actually absorbed. The problem with diabetes, especially T2 diabetes is that elevated insulin levels actually create a false condition that overrides this control system by apparently increasing demand for lipid storage in the adipose tissues (for future use). and it is this mechanism that is responsiible for the Metabolic Syndrome and beer belly characteristic of T2D. If you reduce the insulin in the blood stream either by meds, exercise, or lowering carb intake levels, then it seems that weight gain can be turned into weight loss. Research has shown that it is not simply calories in / calories out which is why most calorie restricted diets fail unless very severe, and why we are seeing that low carb diets are more effective at weight loss.
If you want the science behind this, then you need to research the Endocrine system, since it is not part of the metabolism or digestion process. but the research is quite mind blowing and beyond the comprehension level normally used on this site. I have given up posting links since it is a different language, and quite complex a subject to follow. But the answers are there, not here.
Edit to add: digestion of fat is made by the enzyme Lipase
https://en.wikipedia.org/wiki/Hormone-sensitive_lipase